### Article

Kyungpook Mathematical Journal 2021; 61(3): 495-512

**Published online** September 30, 2021

Copyright © Kyungpook Mathematical Journal.

### Value Distribution of L-functions and a Question of Chung-Chun Yang

Xiao-Min Li^{*}, Qian-Qian Yuan, Hong-Xun Yi

Department of Mathematics, Ocean University of China, Qingdao, Shandong 266100, P. R. China

e-mail : lixiaomin@ouc.edu.cn and yuanqianqian92@163.com

Department of Mathematics, Shandong University, Jinan, Shandong 250199, P. R. China

e-mail : hxyi@sdu.edu.cn

**Received**: February 22, 2017; **Revised**: May 2, 2021; **Accepted**: May 18, 2021

### Abstract

We study the value distribution theory of L-functions and completely resolve a question from Yang [10]. This question is related to *L*-functions sharing three finite values with meromorphic functions. The main result in this paper extends corresponding results from Li [10].

**Keywords**: Nevanlinna theory, Meromorphic functions, L-functions, Shared Values, Uniqueness theorems.

### 1. Introduction and Main Results

Throughout this paper, by meromorphic functions we will always mean meromorphic functions in the complex plane. We assume that the reader is familiar with the basic notions and results in the Nevanlinna theory, which can be found, for example, in [4, 9, 18, 19]. It will be convenient to let

Let

This paper concerns the question of how an L-function is uniquely determined in terms of the pre-images of complex values in the extended complex plane, or sharing values. We refer the reader to the monograph [17] for a detailed discussion on the topic and related works. Throughout the paper, an L-function always means an L-function L in the Selberg class, which includes the Riemann zeta function

(i) Ramanujan hypothesis.

(ii) Analytic continuation. There is a nonnegative integer

(iii) Functional equation.

where

with positive real numbers

_{j},

(iv) Euler product hypothesis.

We first recall the following result due to Steuding [17], which actually holds without the Euler product hypothesis:

**Theorem A.** ([17, p.152]) If two L-functions _{1}_{2}

**Remark 1.1.** In 2016, Hu-Li [6] pointed out that Theorem A is false when _{1}_{2}_{1}

and _{2}

Thus, _{1}_{2}_{1} - 1_{2} - 1

Theorem A implies that two L-functions with

**Theorem B.**([10]) Let

**Remark 1.2.** The number "two" in Theorem B is the best possible, as shown by the above example with

By Theorem B we can get the following result:

**Corollary A.**([10]) Let

In a communication to Professor Li, Yang asked the following question:

**Question A.**([10]) If

**Remark 1.3.** By taking

Next we consider the first, the second and the fourth Painlevé equations given respectively by

In 2007, Lin-Tohge [13] obtained some results similar to Theorem B. Indeed, Lin-Tohge [13] studied some shared-value properties of the first, the second and the fourth Painlevé transcendents by applying their distinctive value distribution, and proved the following results:

**Theorem C.**([13, Theorem 1]) Let ω be an arbitrary nonconstant solution of one of the equations (PI), (PII) and (PIV), and let _{1},_{2},_{3},_{4}_{1},_{2},_{3},_{4}

**Theorem D.**([13, Theorem 2]) Let ω be an arbitrary solution of (PI) and _{1}_{2}_{1}_{2}

**Theorem F.**([13, Theorem 3]) Let ω be an arbitrary solution of (PI). Then there does not exist a pair of two finite values

Regarding Theorem B, one may ask, what can be said about the conclusion of Theorem B if we remove the assumption "

**Theorem 1.1.** Let

By Theorem 1.1 we get the following result:

**Corollary 1.2.** If

As a special case of Corollary 1.2, we give the following result which completely resolves Question A:

**Corollary 1.3.** If

In the same manner as in the proof of Theorem 1.1, we can get the following result by Lemma 2.10 in Section 2 of the present paper:

**Theorem 1.4.** Let ^{}(k)}^{}(k)}

Throughout this paper, we will apply Nevanlinna theory to prove the main result in this paper.

### 2. Preliminaries

In this section, we will give some important lemmas to prove the main result of the present paper. For convenience in stating the following first result from Gundersen [3], we shall use the following notation: we shall let

of distinct pairs of integers that satisfy

**Lemma 2.1.**([3, Corollary 2]) Let

The following result is due to Mokhon-ko [14]:

**Lemma 2.2.**(Valiron-Mokhon-ko lemma, [14]) Let _{k}}_{j}}_{p}≠ 0

We also need the following result due to Lahiri-Sarkar [8]:

**Lemma 2.3.**([8, Lemma 6]) Let

The following result is from Gundersen [2]:

**Lemma 2.4.**([2, Theorem 3]) Suppose that

as

**Lemma 2.5.**([20, proof of Lemma 4]) Let

**Lemma 2.6.**([21, Lemma 6]) Let _{1}_{2}

for _{1}_{2},

For introducing the following result, we first give the following notation (cf.[20]): Let _{0}(r)_{0}(r)_{0}(r).

The following lemma is essentially due to Zhang [21]:

**Lemma 2.7.**([21, proof of Theorem 1 and Theorem 2]) Let

If

and

(i)

(ii)

(iii)

Here γ is a nonconstant entire function,

**Lemma 2.8.** ([22]) Let ^{s}=1,

Finally we prove the following result which plays an important role in proving the main results of this paper:

**Lemma 2.9.** Let

(i)

(ii)

Here

and

By Lemma 2.4 we have

By (2.4), Lemma 2.5 and the assumption of Lemma 2.9 we have

and

By (2.1)-(2.3), (2.5), (2.6) and the assumption that

By (2.1)-(2.3) and the assumption that _{1},_{2}_{0}

and

Set

Then from (2.1), (2.2), (2.3) and (2.10) we can deduce

If

then

where

Again from (2.10) and (2.12) we have

By integrating two sides of (2.14) we can get

where

By (2.8), (2.13) and (2.16) we can get

which together with (2.8) gives

Set

and so we have

By (2.3), (2.11) and (2.19) we deduce

and

By (2.1), (2.3) and (2.9) we have

Thus

On the other hand, by (2.10) and (2.19) we have

By (2.3), (2.5), (2.21), (2.22) and (2.25) we easily deduce

and

By (2.28) and Lemma 2.5 we deduce

By (2.11), (2.20) and (2.26) we deduce

In the same manner as above we get

and

By (2.28), (2.29) and (2.31) we get the conclusion (i) of Lemma 2.9. By (2.27), (2.30) and (2.32) we get (ii) of Lemma 2.9. This completely proves Lemma 2.9.

**Lemma 2.10.**([5]) Let

such that, for every positive integer

### 3. Proof of Theorem 1.1.

First of all, we denote by _{j}

which together with the definition of the order of a meromorphic function implies that

By noting that

On the other hand, by the assumption that

i.e.,

as

as

By (3.4), (3.5), the definition of the order of a meromorphic function and the standard reasoning of removing an exceptional set we deduce

Now we set

By the assumption that

for all

By (3.8) we can see that _{1}.

where

we deduce by (3.1), (3.5) and (3.10) that

as

By (3.9) we consider the following two cases:

**Case 1.** Suppose that there exists a subset

Next we prove

Indeed, if

by (3.9), (3.11), (3.14) and the assumption that

which contradicts (3.12), and so (3.13) is valid. By (3.9) and (3.13) we get the conclusion of Theorem 1.1.

**Case 2.** Suppose that at most there exists a subset

Then, by (3.15) we have

as

Noting the assumption that

We discuss the following two subcases:

**Subcase 2.1.** Suppose that

**Subcase 2.1.1.** Suppose that

Suppose that

where α is a nonconstant entire function. By the right formulae of (3.17) and (3.19) we have

By (3.20) and Lemma 2.2 we have

By (3.6), (3.21) and the definition of the order of a meromorphic function we have _{1}

which contradicts (3.1).

uppose that

**Subcase 2.1.2.** Suppose that

where β is an entire function. By (3.17), (3.22) and Lemma 2.2 we have

By (3.6), (3.23) and the definition of the order of a meromorphic function we have _{4}≠ 0_{2}

which contradicts (3.1).

Suppose that

**Subcase 2.2.** Suppose that

By (2.1)-(2.4), (2.7)-(2.9) and (3.24) we deduce

We consider the following two subcases:

**Subcase 2.2.1.** Suppose that

Then, by (3.25) and (3.26) we have

By (2.7), (3.27) and Lemma 2.6 we know that there exist two integers

By substituting (2.1) and (2.2) into (3.28) we get

By noting that

where γ is a nonconstant entire function, _{0}(r)

By (3.17), (3.30), Lemma 2.2 and Lemma 2.8 we have

By (3.6) and (3.31) we have _{3}

which contradicts (3.1).

**Subcase 2.2.2.** Suppose that

By noting that

where

i.e.,

By (3.34) we deduce that

### Acknowledgements.

The authors wish to express their thanks to the referee for his/her valuable suggestions and comments.

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